Degree of Orthomorphism Polynomials over Finite Fields

An orthomorphism over a finite field \(\mathbb{F}_q\) is a permutation \(\theta:\mathbb{F}_q\mapsto\mathbb{F}_q\) such that the map \(x\mapsto\theta(x)-x\) is also a permutation of \(\mathbb{F}_q\). The degree of an orthomorphism of \(\mathbb{F}_q\), that is, the degree of the associated reduced permutation polynomial, is known to be at most \(q-3\). We show that this upper bound is achieved for all prime powers \(q\notin\{2, 3, 5, 8\}\). We do this by finding two orthomorphisms in each field that differ on only three elements of their domain. Such orthomorphisms can be used to construct \(3\)-homogeneous Latin bitrades.